8. The general solution of a linear system.

Consider the linear dynamical system

 

that is

 

We look for solutions on the form

 

If we instert these to the system we get

 

that is

 

This equation system has nontrivial solutions if and only if

 

that is, if and only if

 

This is the characteristic equation of the system and the solutions  its eigenvalues. If we now for each eigenvalues   solve the equation system

 

we obtain the eigenvectors

 

Then we have the general solution of our original dynamical system as

 

that is

 

where C₁ and C₂ are arbitrary constants.

Example 8: Assume that the matrix A is

 

This matrix has characteristic equation

 

with the eigenvalues 

 

and corresponding eigenvectors

 

This means that the general solution of the corresponding dynamical system is

 

that is

 

Example 9: Solve the system

 

Solution: We consider the matrix

 

and its characteristic equation

 

that is

 

The eigenvalues thus are

 

with corresponding eigenvectors

 

This means that the dynamical system has the general solution

 

that is

 

These are all complex solutions. We are actually only interested in the realsolutions. With help of Euler’s formula we get

If we now pick arbitrary real constants D₁ and D₂ and put

 

we get the general real solution